Your blog entry seems to be targeted specifically at exercises accompanying a lecture, i.e. exercises that students do while studying a new subject. Do you also have some insightful opinion on students that are already more experienced (like revisiting some subject)? So e.g. do you agree with the widespread opinion that one should do every exercise in Atiyah-Macdonald and Hartshorne and… somewhen if one wishes to go in that direction?

LikeLike

]]>LikeLike

]]>LikeLike

]]>For those of us with worse rulers or who just suck at constructing perpendiculars, there’s an easier way to construct the orthocenter. Draw the A altitude and let it intersect BC at D and (ABC) at P. Then draw the circle with center D through P, and its other intersection with AD will be H (the orthocenter). This works because the reflection of H about BC lies on the circumcircle.

LikeLike

]]>LikeLike

]]>This is easier to prove when you prove it converges *absolutely* almost everywhere, and indeed one sees that if the original theorem is true, then so must be this absolute convergence theorem, since if each $f_i$ is replaced by its absolute value we can apply the theorem on the new sequence (the hypothesis is obviously still true) to get new sum function $g(x) = \sum_{i=1}^{\infty} |f_i|(x)$ which is guaranteed to converge almost everywhere; hence $f$ converges absolutely almost everywhere.

LikeLike

]]>LikeLike

]]>LikeLike

]]>LikeLike

]]>LikeLike

]]>